Right-hand rule

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For the related yet different principle relating to electromagnetic coils, see right hand grip rule.

In mathematics and physics, the right-hand rule is a convention for determining relative directions of certain vectors.

[edit] Types

In fact, there are three closely related right-hand rules.

[edit] Direction associated with an ordered pair of directions

The first of these occurs in situations in which a non-commutative operation must be performed on two directions a and b (in a three-dimensional space) that constructs a direction c perpendicular to both a and b. There are in fact two such directions. The right-hand rule imposes the following procedure for choosing one of the two directions.

• First, the hand is held flat and positioned so that the fingers are aligned with a. Then, the hand is rotated about the forearm so that the fingers curl inward toward b. The thumb indicates c.

• There is also an alternative technique. First, the forefinger of the right hand is pointed directly forward, and the entire hand positioned so that the forefinger is aligned with a. Then, the middle finger is turned inward (toward the palm), and the hand is turned about the axis defined by a so that the middle finger aligns with b. The thumb indicates c.

• Alternatively, consider a person standing at the origin facing along a and the direction of their head (in their own frame) being b. Then their right hand gives c

• In electromagnetism (physics), yet a third technique is used for determining the direction of a force due to a positive charge (conventional current) moving in a magnetic field (see Lorentz force). Using the right hand, the fingers represent the direction of the magnetic field, the thumb is the direction of the current, and the resulting force is in the direction the palm appears to "push". For more information, see Right hand grip rule.

[edit] Direction associated with a rotation

The other form of the right-hand rule occurs in situations where a direction c must be determined based on a rotational direction, or vice versa. In this case, the fingers of the right hand are curled in the rotational direction, and the thumb indicates c.

Correspondingly:

• moving away from the observer is associated with clockwise rotation and moving towards the observer with counterclockwise rotation, like a screw
• leftward direction is associated with the rotation of the wheels of a vehicle moving forward

The relation with the previous section is established by associating with directions a and b the rotation through the shorter angle from a to b, clockwise or counterclockwise.

The first form of the rule is also used to determine the direction of the cross product of two vectors. This leads to widespread use in physics, wherever the cross product occurs. A list of physical quantities whose directions are related by the right-hand rule is given below. (Some of these are related to cross products only indirectly, and use the second form.)

• The angular velocity of a rotating object and the rotational velocity of any point on the object
• A torque, the force that causes it, and the position of the point of application of the force
• A magnetic field, the position of the point where it is determined, and the electric current (or change in electric flux) that causes it
• A magnetic field in a coil of wire and the electric current in the wire
• The force of a magnetic field on an object, the magnetic field itself, and the velocity of the object
• The induced current from motion in a magnetic field (known as Fleming's right hand rule)

[edit] Left-hand rule

Fleming's left hand rule is a rule for finding the direction of the thrust on a conductor carrying a current in a magnetic field.

The left-hand rule can also refer to a reversal of the right-hand rule, where one of the vectors is reversed and so creates a left-handed triad instead of a right-handed triad.

An example of this situation is for left-handed materials. Normally, for an electromagnetic wave, the electric and magnetic fields, and the direction of propagation of the wave obey the right-hand rule. However left handed materials have special properties (the negative refractive index), it makes the direction of propagation point in the opposite direction.